I cannot for the life of me figure out what the hell that thing is supposed to be doing. I've seen several CVT designs over the past 30 years or so, and this one appears to be useless. Maybe a better video would change my opinion.

You could see that the design is very similar, the only thing that changes is the situation of the hydraulic system. In the original design these are in the bottom housing, not in the top.
I prefer to put them in the top housing and compare the flector moment and more things(don't matter)
This system work like this:
-The red shaft transmits the torque from the engine(Like an standard gearbox).
-The grey discs receive orders from the ECU and turn some specific angle (done by the engine map).

-Now we have passed the first stage, and our second red shaft is turning with a diferent w(rad/s) and in the opposite direction.

-The second stage is just the same, The second red shaft transmit the power to the last shaft. And the second grey discs change the W (Like you know P=W*T, so if you change W, you change also the torque T).

This design is working in some NISSAN models, is extremely comfortable and soft for driving....but is very expensive and of course has the limitation of friction between discs and shaft.

I cannot for the life of me figure out what the hell that thing is supposed to be doing. I've seen several CVT designs over the past 30 years or so, and this one appears to be useless. Maybe a better video would change my opinion.

I had the same reaction when I saw it. All the CVT's I've seen are belts riding between 2 sets of conical sheaves. This one is very different. There is a main transmission shaft running from bottom left to upper right in the picture and the idea is to have an input shaft rotating at one speed (say the bottom, left in the picture) and the output shaft to rotate at another speed (say the top, right in the picture). You'll notice that the main shaft has two sets of conical things. These two sets are essentially identical. Where the small sections of the conical parts meet, they are actually separate, so the main shaft is in 3 different pieces, all rotating on the same axis.

The four vertical shafts in the picture are each attached to a disk that runs against the conical things. The disks rotate freely and press against the two conical things allowing the one conical thing to drive this disk and the disk is then driving the second conical thing in each set. There are 2 sets.

The picture shown here, shows just 1 of the 2 sets of conical things and discs. Note the 2 conical things are separated at the small end so they can rotate at different speeds.
http://desmond.imageshack.us/Himg339/scaled.php?server=339&filename=cvt2.jpg&res=landing [Broken]
The disk is being rotated so that the edge of the disk touches the one conical thing at one point and the second conical thing at a second point. Where that disk touches each conical thing in the picture is indicated by the red stripe on the conical thing. By rotating the vertical shaft, this disk touches the two conical things at different locations, acting as a 'gear' of sorts so the two conical things rotate at different speeds.

cvt-extroid, I think with anything that's new like this, you won't find much documentation, if any. You have to reduce it to basic principals. There is a certain geometry to the transmission that dictates its performance, but to determine if the thing can hold together, you need to look at forces, stresses, frictional loads, etc... There's no cook book that will guide you through the design of something like this, you need to apply basic principals of engineering to determine how to design it. Also, that won't do much for you when it comes to wear rates of things like the disk riding on the two conical things. What materials to use to minimize wear is one of those 'black art' issues that you can only get a handle on by building and running one of these things. And it's those kinds of black art features that are often the most difficult to work out the details on.

Personally, I'd like to see a transmission like this built for industry to do away with VFD motors and the controls associated with them. Everyone wants to go after the automotive market but there are plenty of other applications for this type of technology and it seems no one is going after it.

Think so. It's extremely difficult to find some good information about this CVT.
And like you say i developed some basics analysis about contact stresses,ratio of transmision,fatigue...things that you can study in a meng(mechanical engineering) basic course.
To improve the design with the proper materials, lubricants,study the effects of the friction damage.... would be necessary years for that.

Here more info about how that it works. Detail of the differents transmission ratio.
Rápido=fast.
Lento=slow.
You can see that the slow disc has a bigger Torque than the fast disc, to keep the relation P=T*W.
http://desmond.imageshack.us/Himg140/scaled.php?server=140&filename=cvttoroidalcopy.jpg&res=landing [Broken]

Thinking about it some more, this gearbox has an inherent weakness. Where the cone and disk touch is essentially a single point of contact. Sure, the two will flatten out slightly at this point, but the area of contact won't be very large unless at least one of the two parts has a very low modulus (ie: is very rubbery). But as the contact point increases, other problems are created.

The contact stress between the parts is necessarily much higher than a belt and V sheave type arrangement of a more conventional CVT of the same size. You could compare this with surface hardened steel on steel with oil lubrication such as used in ball bearings for example. Bearing surfaces fatigue quickly unless the contact stress is very low. Take a look at bearing contact stresses and fatigue. I'd be very skeptical that this could be made to work reliably.

Yes, contact stresses is the main weakness. It's just the same thing the teacher said when i asked him about this work. And for me honestly i think is a challengue to find the proper material.
I tried some numbers with the geometry i designed and i supposed using steel and that's what i got.
First we take this section:
http://desmond.imageshack.us/Himg525/scaled.php?server=525&filename=contacto1.jpg&res=landing [Broken]
where R1=32.1 (cone curvature radious in mm (aprox)) and R2=32 (shaft disc curvature).